Quantum Algorithms for the Maximum K-Plex Problem

The k-plex model, which allows each vertex to miss connections with up to
$k$
neighbors, serves as a relaxation of the clique model. Its adaptability makes it more suitable for analyzing graphs from real-world applications, where noise and imperfect data are common and the stringent clique model is often impractical. The challenge of identifying maximum k-plex (MKP, an NP-hard problem) is gaining attention in fields such as social network analysis, community detection, terrorist network identification, and graph clustering. Recent research efforts have focused on optimizing the time complexity of MKP algorithms. The state-of-the-art has reduced the complexity from a trivial
$O^{*}(2^{n})$
to
$O^{*}(c_{k}^{n})$
, with
$c_{k} > 1.94$
for
$k$
> 3, where
$n$
denotes the number of vertices. In this paper, we demonstrate that MKP can be solved in
$O^{*}(1.42^{n})$
and propose the first two quantum algorithms, qTKP and qMKP, to achieve this complexity. qTKP employs quantum search integrated with graph encoding, degree count, degree comparison, and size determination to find a k-plex of a given size; qMKP uses a binary search to progressively identify the maximum solution. To validate the practical performance and effectiveness of our algorithms, proof-of-principle experiments were conducted using the latest IBM quantum simulator currently available. This work holds potential to be applied to a wide range of clique relaxations, e.g., n-clan and n-club.